Abstract

A well-known result, due to Ostrowski, states that if $\Vert P-Q
\Vert_2< \varepsilon$, then the roots $(x_j)$ of $P$ and $(y_j)$ of
$Q$ satisfy $|x_j -y_j|\le C n \varepsilon^{1/n}$, where $n$ is the
degree of $P$ and $Q$. Though there are cases where this estimate
is sharp, it can still be made more precise in general, in two
ways: first by using Bombieri's norm instead of the classical $l_1$
or $l_2$ norms, and second by taking into account the multiplicity
of each root. For instance, if $x$ is a simple root of $P$, we show
that $|x-y|< C \varepsilon$ instead of $\varepsilon^{1/n}$. The
proof uses the properties of Bombieri's scalar product and Walsh
Contraction Principle.